Geodesic
Some questions on the distribution of zeros of entire functions of several variables
Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 363-380 
L. I. Ronkin. Some questions on the distribution of zeros of entire functions of several variables. Sbornik. Mathematics, Tome 16 (1972) no. 3, pp. 363-380. http://geodesic.mathdoc.fr/item/SM_1972_16_3_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this article the idea of the $\Gamma$-capacity of a set in $C^n$, the analog of the idea of capacity of a set in $C^1$, is introduced. The basic result of the paper (Theorems 2 and 3) is the following: if the function $f(z,\omega)$, where $z\in C^n$, and $\omega\in C^1$, has only a finite number of zeros as a function of $\omega$ for all $z$ in some set of positive $\Gamma$-capacity, then it is the product of an entire pseudopolynomial in $\omega$ and an entire function which is never zero. Bibliography: 12 titles.

[1] G. Julia, “Sur le domaine d'existence d'une fonction implicite definie par une relation entiere $G(x,y)=0$”, Bull. Soc. Math. France, 54 (1926), 26–28 | MR

[2] P. Lelong, “Sur quelques problémes de la theorie des fonctions de deux variables complexes”, Ann. Scient. Ecole Norm. Super., 58 (1941) | MR | Zbl

[3] L. I. Ronkin, “O regulyarizatsii supremuma semeistva plyurisubgarmonicheskikh funktsii i ee primenenii k analiticheskim funktsiyam mnogikh peremennykh”, Matem. sb., 71(113):1 (1966), 131–142 | MR | Zbl

[4] L. I. Ronkin, “O roste plyurisubgarmonicheekikh funktsii i o raspredelenii znachenii tselykh funktsii mnogikh peremennykh”, DAN SSSR, 179:2 (1968), 290–292 | MR | Zbl

[5] L. I. Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, Nauka, Moskva, 1971 | MR | Zbl

[6] L. I. Ronkin, “Ob analoge kanonicheskogo proizvedeniya Veiershtrassa dlya tselykh funktsii mnogikh peremennykh”, Trudy Mosk. matem. ob-va, 18 (1968), 105–146 | MR | Zbl

[7] L. I. Ronkin, “O roste tselykh funktsii mnogikh kompleksnykh peremennykh”, Matem. sb., 71(113) (1966), 337–356 | MR

[8] S. Bokhner, U. T. Martin, Funktsii mnogikh kompleksnykh peremennykh, IL, Moskva, 1951

[9] M. Erve, Funktsii mnogikh kompleksnykh peremennykh, Mir, Moskva, 1965 | MR

[10] P. Lelong, “Fonctions entieres ($n$ variables) et fonctions plurisousharmoniques d'ordre fini dans $C^n$”, J. Analyse Math., 12 (1965), 365–407 | DOI | MR

[11] W. Stoll, “Ganze Functionen endlicher Ordnung mit gegebenen Nullstellenflachen”, Math. Z., 57 (1963), 211–237 | DOI | MR

[12] G. Rémoundos, “Extension aux fonctions algebroides multiformes du theoreme de M. Picard et de ses generalisations”, Mem. Scient. Math. Paris, no. 23, 1927